Orthogonal Reverse Derivations on semiprime Γ-semirings

Year 2019, Volume: 7 Issue: 1, 71 - 77, 30.04.2019

B. Venkateswarlu , M. Murali Krishna Rao , Y. Adi Narayana

https://doi.org/10.36753/mathenot.559255

Cited By: 1

Abstract

In this paper, we introduce the notion of reverse derivation and orthogonal reverse derivations on

Γ-semirings. Some characterizations of semi prime Γ-semirings are obtained by means of orthogonal

reverse derivations. And also obtained necessary and sufficient conditions for two reverse derivations to

be orthogonal.

Keywords

Γ-semirings , semi prime , derivations , orthogonal reverse derivations

References

• [1] Allen, P.J., A fundamental theorem of homomorphism for semirings, Proc. Amer. Math. Soc., 21 (1969), 412-416.
• [2] Bresar, M. and Vukman, J., On the left derivation and related mappings, Proc. Amer. Math. Soc., 10 (1990), 7-16.
• [3] Dutta, T.K. and Kar, S., On regular ternary semirings, Advances in Algebra, Proceedings of the ICM Satellite Conference in Algebra and Related Topics,World Scientific, (2003) 343-355.
• [4] Javed, M.A., Aslam, M. and Hussain, M., On derivations of prime Γ-semirings, Southeast Asian Bull. of Math., 37 (2013), 859-865.
• [5] Lehmer, H., A ternary analogue of abelian groups, Amer. J. of Math., 59 (1932), 329-338.
• [6] Lister, W.G., Ternary rings, Tran. of Amer. Math. Soc., 154 (1971), 37-55.
• [7] Murali Krishna Rao, M., Γ-semirings-I, Southeast Asian Bull. of Math., 19(1) (1995), 49-54.
• [8] Murali Krishna Rao, M., Γ-semirings-II, Southeast Asian Bull. of Math., 21 (1997), 281-287.
• [9] Murali Krishna Rao, M. and Venkateswarlu, B., Regular Γ-semirings and field Γ-semirings, Novi Sad J. of Math., 45 (2) (2015), 155-171.
• [10] Neumann, V., On regular rings, Proc. Nat. Acad. Sci., 22 (1936), 707-13.
• [11] Nobusawa, N., On a generalization of the ring theory, Osaka. J. Math., 1 (1964), 81 - 89.
• [12] Posner, E.C., Derivations in prime rings, Proc. Amer. Math. Soc., 8 (1957), 1093-1100.
• [13] Sen, M.K., On Γ-semigroup, Proc. of International Conference of algebra and its application, (1981), Decker Publicaiton, New York, 301-308.
• [14] Suganthameena, N. and Chandramouleeswaran, M., Orthogonal derivations on semirings, Int. J. Of Cont. Math. Sci., 9 (2014), 645-651.
• [15] Vandiver, H.S., Note on a simple type of algebra in which the cancellation law of addition does not hold, Bull. Amer. Math., 40 (1934), 914-921.